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Aims/ objectives : To obtain the analytical solutions of the governing fourth order partial differential equations with variable and singular coefficients of non-uniform elastic beams under constant and harmonic variable loads travelling at varying velocity.
Study design: The study makes use of the governing equation of beam incorporating some parameters.
Place and Duration of Study: Department of Mathematical Sciences, Adekunle Ajasin University, P.M.B 01, Akungba-Akoko, Nigeria, Federal University of Technology, Akure, Nigeria, between July
2016 and July 2017.
Methodology: The governing equation of the problem is a fourth order partial differential equation. In order to solve this problem, elegant technique called Galerkin’s Method is used to reduce the
governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations.

Results: The results show that response amplitudes of the non uniform beam decrease as the value of the axial force N increases. Furthermore, for fixed value of axial force N, the displacements of the simply supported non uniform beam resting on elastic foundations decrease as the foundation modulus K increases. The results further show that, for fixed N and K, it is observed that higher values of the load longitudinal frequency produce more stabilizing effects on the elastic beam.
Conclusion: Higher values of axial force N and foundation moduli K reduce the risk factor of resonance in a vibrating system. Also higher load longitudinal frequency produce more stabilizing effects on the elastic beam thereby reduce resonance in a vibrating system.